Supernova Neutrinos: the Role of the Nuclear Equation of State and Neutrino-Matter Interactions in Hot and Dense Matter

Supernova Neutrinos: The role of the nuclear equation of state and neutrino-matter interactions in hot and dense matter

• Introduction • Temporal structure: Neutron star tomography • Neutrino spectra • Supernova cooling constraints

Sanjay Reddy INT, Univ. of Washington, Seattle

SN neutrinos at DUNE: SLAC meeting, Nov. 23-24, 2015 Supernova Neutrinos 1500 km

Core collapse

tcollapse ~100 ms neutrinos diffuse 3X107 km out of the dense Shock wave newly born E ~1051ergs 10 km shock neutron star carry away 53 Quasi-static ~ 3 x 10 ergs ~ 1 s 100 km • The time structure of the neutrino signal depends on how heat is transported in the neutron star core (1013-1015 g/cm3 ). • The spectrum is set by scattering in a hot (T=5-10 MeV) and not so dense (1011-1013 g/cm3 ) neutrino-sphere. A broad brush description of SN neutrino emission Neutrino luminosity:

L (t) 4⇡ R2 (t) T 4(t) ⌫ ⇡ ⌫ ⌫ ⌫

Radius of the neutrino Temperature of the neutrino decoupling surface. decoupling surface.

For given neutron star mass:

• The equation of state determines R. • Neutrino interactions near the PNS surface determines T. • Time evolution is determined by neutrino opacities in the core, equation of state of dense matter and the stability of the ﬁnal neutron star. 34 Madappa Prakash et al. Neutrino Emission: Baseline Model

Generic expectations:

• For a galactic SN we should see neutrinos for about 40 s

• Luminosity increases with NS mass.

• Heavier masses can collapse to a black-hole if the high density equation of state is soft.

Prakash, Lattimer, Pons, Steiner and Reddy, astro-ph/0012136

Fig. 14. The evolution of the total neutrino luminosity for stars of various baryon masses. Shaded bands illustrate the limiting luminosities corresponding to a count rate of 0.2 Hz in all detectors, assuming a supernova distanceof50kpcforIMBand Kamioka, and 8.5 kpc for SNO and SuperK. The width of the shadedregionsrepresents uncertainties in the average neutrino energy from the use of adiﬀusion scheme for neutrino transport

5.4 What Can We Learn From Neutrino Detections? The calculations of Pons, et al. [73] show that the variationsintheneutrino light curves caused by the appearance of a kaon condensate in astablestarare small, and are apparently insensitive to large variations intheopacitiesassumed for them. Relative to a star containing only nucleons, the expected signal differs by an amount that is easily masked by an assumed PNS mass difference of 0.01 0.02 M⊙.Thisisinspiteofthefactthat,insomecases,afirstorder phase− transition appears at the star’s center. The manifestations of this phase transition are minimized because of the long neutrino diffusion times in the star’s core and the Gibbs’ character of the transition. Both act in tandem to prevent either a “core-quake” or a secondary neutrino burst from occurring during the Kelvin-Helmholtz epoch. Observable signals of kaon condensation occur only in the case of metastable stars that collapse to a black hole. In this case, the neutrinosignalforastar closer than about 10 kpc is expected to suddenly stop at a levelwellabovethat of the background in a sufficiently massive detector with a low energy threshold such as SuperK. This is in contrast to the signal for a normal star of similar mass for which the signal continues to fall until it is obscured by the background. The lifetime of kaon-condensed metastable stars has a relatively small range, of order Protoneutron Star Evolution

Initial state of the PNS. Pons et al. (1999) Neutrino diffusion cools and deleptonizes the PNS. Time scales are set by properties of high density matter

Typical time-scales:

% t ( T(t) ≈ T(t = 0) '1 − * & τC ) R2 τC ≈ CV c λν

3 µ Y (t) ≈ ν ν 6π 2 % t ( ≈ Y(t = 0) exp' − * & τD ) 3 ∂Y R2 τ = L D 2 Y c π ∂ ν λνe

Pons et al. (1999)

€ Neutrino diffusion cools and deleptonizes the PNS. Time scales are set by properties of high density matter

Typical time-scales:

% t ( T(t) ≈ T(t = 0) '1 − * & τC ) R2 τC ≈ CV c λν

3 µ Y (t) ≈ ν ν 6π 2 % t ( ≈ Y(t = 0) exp' − * & τD ) 3 ∂Y R2 τ = L D 2 Y c π ∂ ν λνe

Pons et al. (1999)

€ Neutrino Transport

• RHS of the Boltzmann Equation.

@f(E ) d3k 1 = 3 R(E ,E , cos ✓) f (1 f ) @t (2⇡)3 1 3 3 1 Z R(E ,E , cos ✓) f (1 f ) 3 1 1 3 +R(E , E , cos ✓)(1 f )(1 f ) 1 3 1 3 R( E ,E , cos ✓) f f 1 3 1 3

q = E E 0 1 3

Dense Matter q = E + E q0 = E1 E3 0 1 3 Neutrino Transport

• RHS of the Boltzmann Equation.

@f(E ) d3k 1 = 3 R(E ,E , cos ✓) f (1 f ) @t (2⇡)3 1 3 3 1 scattering-in Z R(E ,E , cos ✓) f (1 f ) 3 1 1 3 scattering-out +R(E , E , cos ✓)(1 f )(1 f ) 1 3 1 3 pair-production R( E ,E , cos ✓) f f 1 3 1 3 pair-annihilation

q = E E 0 1 3

Dense Matter q = E + E q0 = E1 E3 0 1 3 Neutrino Transport

• RHS of the Boltzmann Equation.

q = E E 0 1 3 pair ! Dense | | Matter scattering

q0 = E1 + E3 q0 = E1 E3 q Neutrino Cross Sections

Differential Scattering/Absorption Rate: response function of the medium

Neutrino Opacities in Nuclear2 Matter 13 R(E1,E3, cos ✓)=GF L(E1,E3, cos ✓) S[⇢,Ye,T ](q0,q) 5. Inelastic Neutrino Interactions with⇥ Relativistic Nucleons and Leptons neutrino/lepton kinematic factor We now explore more sophisticated formalisms for handling inelastic scattering processes in nuclear matter. In this section, we address the • Neutralnon-interacting and charged nucleon current case. reactions contribute. For neutrino energies of interest to supernova, which are less than afewhundredMeV,wemaywritetheneutrino-matterinteraction in terms of Fermi’s eﬀective Lagrangian

cc GF µ = lµj for νl + B2 l + B4 (47) Lint √2 W → nc GF ν µ = l j for νl + B2 νl + B4 , (48) Lint √2 µ Z →

−49 −3 where GF 1.436 10 erg cm is the Fermi weak coupling constant. When the≃ typical× energy and momentum involved in the reactionare small (compared to the mass and 1/size of the target particle)thelepton and baryon weak charged currents are:

l = ψ γ (1 γ ) ψ ,jµ = ψ γµ (g g γ ) ψ . (49) µ l µ − 5 ν W 4 V − A 5 2 Similarly, the baryon and neutrino neutral currents are given by

lν = ψ γ (1 γ ) ψ ,jµ = ψ γµ (c c γ ) ψ , (50) µ ν µ − 5 ν Z 4 V − A 5 2 where 2 and 4 are the baryon (or electron) initial state and ﬁnal state labels, respectively (these are identical for neutral-current reactions). The vector and axial-vector coupling constants (cV ,gV & cA,gA)are listed in Table 1 for the various charged- and neutral-current reactions of interest. The charged-current reactions are kinematicallysuppressedfor

νµ and ντ neutrinos. This is because their energy Eνµ/ντ T mµ,mτ . On the other hand, neutral-current reactions are common≃ to a≤ll neutrino species and the neutrino-baryon couplings are independent of neutrino flavor. Neutrino coupling to the lepton in the same family is modified since the scattering may proceed due to both W and Z exchange; the couplings shown in Table 1 reflect this fact. From the structure of the current-current Lagrangian, we cancalcu- late the differential cross section for neutrino scattering and absorption. We are generally interested in calculating scattering/absorption rates in matter. Hence, it is convenient to express results in terms ofthediffer- ential scattering/absorption rate. For a neutrino with energy E1,thisis given by dΓ(E1)= i cdσ(E1)i/V ,wheredσ is the differential cross section, the sum is over the target particle in volume V ,andc =1isthe ! Neutrino Opacities in Nuclear Matter 13 5. Inelastic Neutrino Interactions with Relativistic Nucleons and Leptons We now explore more sophisticated formalisms for handling inelastic Neutrino OpacitiesNeutrino in Nuclear Matter Cross Sections 13 scattering processes in nuclear matter. In this section, we address the Differential5.non-interacting Scattering/Absorption Inelastic nucleon Neutrino case. Rate: Interactions with For neutrinoRelativistic energies Nucleons of interest to and supernova, Leptons which are less than afewhundredMeV,wemaywritetheneutrino-matterinteraction in termsWe ofnow Fermi’s explore eff moreective sophisticated Lagrangian response formalisms forfunction handling iofnelastic the medium scattering processes in nuclear matter. In this section, we address the non-interactingcc nucleonGF case.µ Neutrino Opacitiesint in= Nuclearlµ Matterj for νl + B2 l + B4 (47)13 For neutrinoL energies√22 ofW interest to supernova,→ which are less than R(E1,E3, cos ✓)=GF L(E1,E3, cos ✓) S[⇢,Ye,T ](q0,q) afewhundredMeV,wemaywritetheneutrino-matterinteractnc GF ν µ ⇥ ion in 5. Inelasticint = NeutrinolµjZ Interactionsfor νl + B2 withνl + B4 , (48) termsRelativistic of Fermi’sL effective√2 Nucleons Lagrangian and Leptons→ neutrino/leptonGF−49 kinematic−3 factor Wewhere nowGF explorecc1.436 more10 sophisticatedergµ cm is formalisms the Fermi weak for handling coupling constant. inelastic ≃int = × lµjW for νl + B2 l + B4 (47) scatteringWhen the processesL typical in energy√ nuclear2 and matter. momentum In this involved section,→ in the we address reactionare the • non-interactingsmall (comparednc nucleon to theGF case. massν µ and 1/size of the target particle)thelepton Neutral and chargedint = currentlµj Zreactionsfor contribute.νl + B2 νl + B4 , (48) Forand neutrino baryonL weak energies charged√2 of interest currents to are: supernova,→ which are less than afewhundredMeV,wemaywritetheneutrino-matterinteract−49 −µ3 µ ion in where GlFµ = 1ψ.l436γµ (110 γ5) ergψν ,j cm W is= theψ4γ Fermi(gV weakgAγ coupling5) ψ2 . constant.(49) termsWhen of Fermi’s the≃ typical effective× energy− Lagrangian and momentum involved− in the reactionare smallSimilarly, (compared the baryon to the and mass neutrino and 1/size neutral of the currents target are particle given)thelepton by cc GF µ and baryon weak= chargedlµj currentsfor are: νl + B2 l + B4 (47) Lintν √ W µ µ → lµ = ψνγµ (12 γ5) ψν ,jZ = ψ4γ (cV cAγ5) ψ2 , (50) G − µ µ − lµnc= ψlγµ (1F γν5) µψν ,jW = ψ4γ (gV gAγ5) ψ2 . (49) where 2int and= 4 are the− l baryonµj (orfor electron)νl + initialB2− stateνl + andB4 , final state(48) L √2 Z → Similarly,labels, respectively the baryon (these and neutrino are identical neutral for currents neutral-cur are givrenten reactions). by whereTheG vector1.436 and axial-vector10−49 erg cm coupling−3 is the constants Fermi weak (cV coupling,gV & cA constant.,gA)are F ≃lν = ψ ×γ (1 γ ) ψ ,jµ = ψ γµ (c c γ ) ψ , (50) Whenlisted the in typical Tableµ ν 1 energy forµ the− various and5 ν momentum charged-Z 4 and involved neutral-curreV − A in5 the2nt reactio reactionsnare of interest. The charged-current reactions are kinematicallysuppressedfor smallwhere (compared 2 and 4 to are the the mass baryon and (or 1/size electron) of the initial target state particle and)thelepton final state ν and ν neutrinos. This is because their energy E T m ,m . andlabels, baryonµ respectivelyτ weak charged (these currents are identical are: for neutral-curνµ/ντ rent reactions).µ τ On the other hand, neutral-current reactions are common≃ to a≤ll neutrino The vector and axial-vector coupling constants (c ,g & c ,g )are species and the neutrino-baryonµ couplingsµ are independentV V ofA neutrinoA listedlµ in= Tableψlγµ 1(1 for theγ5) variousψν ,j charged-W = ψ4γ and(gV neutral-curregAγ5) ψ2nt. reactions(49) of flavor. Neutrino coupling− to the lepton in the− same family is modified interest. The charged-current reactions are kinematicallysuppressedfor Similarly, the baryon and neutrino neutral currents are given by νsinceand theν scatteringneutrinos. may This proceed is because due their to both energyWEand Z exchange;T m ,m the. couplingsµ τ shown in Table 1 reflect this fact. νµ/ντ ≃ ≤ µ τ On theν other hand, neutral-currentµ reactionsµ are common to all neutrino l = ψ γµ (1 γ5) ψν ,j= ψ γ (cV cAγ5) ψ2 , (50) speciesFromµ and theν the structure neutrino-baryon− of the current-currentZ couplings4 are Lagrangian,− independent weof ca neutrinoncalcu- late the differential cross section for neutrino scattering and absorption. whereflavor. 2 and Neutrino 4 are the coupling baryon to (or the electron) lepton in initial the same state family and is fin malodified state We are generally interested in calculating scattering/absorption rates in labels,since respectively the scattering (these may are proceed identical due to for both neutral-curW and Zrentexchange; reactions). the couplingsmatter. Hence, shown it in is Table convenient 1 reflect to expressthis fact. results in terms ofthediffer- The vector and axial-vector coupling constants (cV ,gV & cA,gA)are ential scattering/absorption rate. For a neutrino with energy E1,thisis listedFrom in Table the 1 structure for the various of the charged-current-current and neutral-curre Lagrangian,nt we reactions cancalcu- of lategiven the by didffΓerential(E1)= crossi cd sectionσ(E1) fori/V neutrino,whered scatteringσ is the diandfferential absorption. cross interest.section, The the charged-current sum is over the target reactions particle are kinematicall in volume V ysuppressedfor,andc =1isthe ν andWe areν neutrinos. generally interested This! is because in calculating their energy scattering/absE orptionT m rates,m in. µ matter.τ Hence, it is convenient to express results inνµ/ termsντ ofthediµ ffer-τ On the other hand, neutral-current reactions are common≃ to a≤ll neutrino ential scattering/absorption rate. For a neutrino with energy E ,thisis species and the neutrino-baryon couplings are independent of neutrino1 given by dΓ(E1)= i cdσ(E1)i/V ,wheredσ is the differential cross flavor.section, Neutrino the sum coupling is over the to the target lepton particle in the in volume same familyV ,and isc =1isthe modified since the scattering may! proceed due to both W and Z exchange; the couplings shown in Table 1 reflect this fact. From the structure of the current-current Lagrangian, we cancalcu- late the differential cross section for neutrino scattering and absorption. We are generally interested in calculating scattering/absorption rates in matter. Hence, it is convenient to express results in terms ofthediffer- ential scattering/absorption rate. For a neutrino with energy E1,thisis given by dΓ(E1)= i cdσ(E1)i/V ,wheredσ is the differential cross section, the sum is over the target particle in volume V ,andc =1isthe ! Neutrino Opacities in Nuclear Matter 13 5. Inelastic Neutrino Interactions with Relativistic Nucleons and Leptons We now explore more sophisticated formalisms for handling inelastic Neutrino OpacitiesNeutrino in Nuclear Matter Cross Sections 13 scattering processes in nuclear matter. In this section, we address the Differential5.non-interacting Scattering/Absorption Inelastic nucleon Neutrino case. Rate: Interactions with For neutrinoRelativistic energies Nucleons of interest to and supernova, Leptons which are less than afewhundredMeV,wemaywritetheneutrino-matterinteraction in termsWe ofnow Fermi’s explore eff moreective sophisticated Lagrangian response formalisms forfunction handling iofnelastic the medium scattering processes in nuclear matter. In this section, we address the non-interactingcc nucleonGF case.µ Neutrino Opacitiesint in= Nuclearlµ Matterj for νl + B2 l + B4 (47)13 For neutrinoL energies√22 ofW interest to supernova,→ which are less than R(E1,E3, cos ✓)=GF L(E1,E3, cos ✓) S[⇢,Ye,T ](q0,q) afewhundredMeV,wemaywritetheneutrino-matterinteractnc GF ν µ ⇥ ion in 5. Inelasticint = NeutrinolµjZ Interactionsfor νl + B2 withνl + B4 , (48) termsRelativistic of Fermi’sL effective√2 Nucleons Lagrangian and Leptons→ neutrino/leptonGF−49 kinematic−3 factor Wewhere nowGF explorecc1.436 more10 sophisticatedergµ cm is formalisms the Fermi weak for handling coupling constant. inelastic ≃int = × lµjW for νl + B2 l + B4 (47) scatteringWhen the processesL typical in energy√ nuclear2 and matter. momentum In this involved section,→ in the we address reactionare the • non-interactingsmall (comparednc nucleon to theGF case. massν µ and 1/size of the target particle)thelepton Neutral and chargedint = currentlµj Zreactionsfor contribute.νl + B2 νl + B4 , (48) Forand neutrino baryonL weak energies charged√2 of interest currents to are: supernova,→ which are less than afewhundredMeV,wemaywritetheneutrino-matterinteract−49 −µ3 µ ion in where GlFµ = 1ψ.l436γµ (110 γ5) ergψν ,j cm W is= theψ4γ Fermi(gV weakgAγ coupling5) ψ2 . constant.gV =1(49),gA =1.26 termsWhen of Fermi’s the≃ typical effective× energy− Lagrangian and momentum involved− in the reactionare smallSimilarly, (compared the baryon to the and mass neutrino and 1/size neutral of the currents target are particle given)thelepton by cc GF µ and baryon weak= chargedlµj currentsfor are: νl + B2 l + B4 (47) Lintν √ W µ µ → lµ = ψνγµ (12 γ5) ψν ,jZ = ψ4γ (cV cAγ5) ψ2 , (50) − µ µ − l = ψ γ G(1F γ ) µψ ,j= ψ γ (g g γ ) ψ . (49) µnc =l µ lν5 j ν Wfor 4 ν + VB Aν5 + B2 , (48) where 2int and 4 are the− baryonµ Z (or electron)l initial2− statel and4 finaln state n L √2 → cV = 1.0,cA = 1.26( 1.1) Similarly,labels, respectively the baryon (these and neutrino are identical neutral for currents neutral-cur are givrenten reactions). by p p c =0.07,c =1.26(1.4) −49 −3 V A whereTheG vector1.436 and axial-vector10 erg cm couplingis the constants Fermi weak (cV coupling,gV & cA constant.,geA)are e F ν µ µ cV =1.92,cA =1. [⌫e] listed in≃l Tableµ = ψν 1×γ forµ (1 theγ various5) ψν ,j charged-Z = ψ4 andγ ( neutral-currecV cAγ5) ψ2nt, reactions(50) of When the typical energy− and momentum involved− in the reactioce nare= 0.08,ce = 1. [⌫ ] interest. The charged-current reactions are kinematicallysuppressedforV A X smallwhere (compared 2 and 4 to are the the mass baryon and (or 1/size electron) of the initial target state particle and)thelepton final state ν and ν neutrinos. This is because their energy E T m ,m . andlabels, baryonµ respectivelyτ weak charged (these currents are identical are: for neutral-curνµ/ντ rent reactions).µ τ On the other hand, neutral-current reactions are common≃ to a≤ll neutrino The vector and axial-vector coupling constants (c ,g & c ,g )are species and the neutrino-baryonµ couplingsµ are independentV V ofA neutrinoA listedlµ in= Tableψlγµ 1(1 for theγ5) variousψν ,j charged-W = ψ4γ and(gV neutral-curregAγ5) ψ2nt. reactions(49) of flavor. Neutrino coupling− to the lepton in the− same family is modified interest. The charged-current reactions are kinematicallysuppressedfor Similarly, the baryon and neutrino neutral currents are given by νsinceand theν scatteringneutrinos. may This proceed is because due their to both energyWEand Z exchange;T m ,m the. couplingsµ τ shown in Table 1 reflect this fact. νµ/ντ ≃ ≤ µ τ On theν other hand, neutral-currentµ reactionsµ are common to all neutrino l = ψ γµ (1 γ5) ψν ,j= ψ γ (cV cAγ5) ψ2 , (50) speciesFromµ and theν the structure neutrino-baryon− of the current-currentZ couplings4 are Lagrangian,− independent weof ca neutrinoncalcu- late the differential cross section for neutrino scattering and absorption. whereflavor. 2 and Neutrino 4 are the coupling baryon to (or the electron) lepton in initial the same state family and is fin malodified state We are generally interested in calculating scattering/absorption rates in labels,since respectively the scattering (these may are proceed identical due to for both neutral-curW and Zrentexchange; reactions). the couplingsmatter. Hence, shown it in is Table convenient 1 reflect to expressthis fact. results in terms ofthediffer- The vector and axial-vector coupling constants (cV ,gV & cA,gA)are ential scattering/absorption rate. For a neutrino with energy E1,thisis listedFrom in Table the 1 structure for the various of the charged-current-current and neutral-curre Lagrangian,nt we reactions cancalcu- of lategiven the by didffΓerential(E1)= crossi cd sectionσ(E1) fori/V neutrino,whered scatteringσ is the diandfferential absorption. cross interest.section, The the charged-current sum is over the target reactions particle are kinematicall in volume V ysuppressedfor,andc =1isthe ν andWe areν neutrinos. generally interested This! is because in calculating their energy scattering/absE orptionT m rates,m in. µ matter.τ Hence, it is convenient to express results inνµ/ termsντ ofthediµ ffer-τ On the other hand, neutral-current reactions are common≃ to a≤ll neutrino ential scattering/absorption rate. For a neutrino with energy E ,thisis species and the neutrino-baryon couplings are independent of neutrino1 given by dΓ(E1)= i cdσ(E1)i/V ,wheredσ is the differential cross flavor.section, Neutrino the sum coupling is over the to the target lepton particle in the in volume same familyV ,and isc =1isthe modified since the scattering may! proceed due to both W and Z exchange; the couplings shown in Table 1 reflect this fact. From the structure of the current-current Lagrangian, we cancalcu- late the differential cross section for neutrino scattering and absorption. We are generally interested in calculating scattering/absorption rates in matter. Hence, it is convenient to express results in terms ofthediffer- ential scattering/absorption rate. For a neutrino with energy E1,thisis given by dΓ(E1)= i cdσ(E1)i/V ,wheredσ is the differential cross section, the sum is over the target particle in volume V ,andc =1isthe ! 3

the symmetry energy is non-linear in the density and large at 100 sub-nuclear density. The electron chemical potential (dashed lines) and neutron proton potential energy difference (solid lines) for these two 10 models are shown as a function of density in beta-equilibrium MeV in ﬁgure 1. Here Y = 0 as a function of density with an as- ∆U sumed temperature of 8 MeV. At sub-nuclear densities, the µ =µ -µ e n p IU-FSU U is always larger than the GM3 U value due to 1/τ 1 σ the larger sub-nuclear density symmetry energy in the former. 0.1 The electron chemical potential as a function of density, as Response of Interacting System well as the equilibrium electron fraction, is shown in ﬁgure

e 1 for both models. In beta-equilibrium, models with a larger

Y 0.05 " " symmetry energy predict a larger electron fraction for a given emission temperature and density and therefore a larger electron chem- 0 0.001 0.01T 0.1 ical potential. Therefore, IU-FSU has a larger equilibrium µe !,q -3 n (fm ) than GM3 and e + n e + p will experience relatively more ! =qB ⇥ ! final state blocking. However, as we show below, the inclusion FIG. 1: Top Panel: The electron chemicalscattering potential (dashed lines) of U in the reaction kinematics is needed for consistency. and U = Un Up (solid lines) are shown as a function of density To elucidate the effects of U we set M⇤ = M and note for the two equation of state models (IUFSU: red curves and GM3: that this assumption can easily be relaxed [1] and it does not Y T change the qualitative discussion below. Because in current black curves) in beta-equilibrium for q⌫ = 0 and = 8 MeV. The grey bandR shows an approximate range of values2 for⇥ inverse spin1 equation of state models the potential, Ui, is independent of relaxationNeutrino time calculated Interactions in [8] and is discussedq = inat connection High> with Densitythe momentum, k, this form of the dispersion relation results collisional broadening. Bottom Panel: The equilibrium electron frac- At small q0 and q theR neutrinoin a free cannot Fermi gas distribution function with single particle tion as a function of density for the two equations of state shown in energies K(k) for nucleons of species i, but with an effective the top panel. resolve a single nucleon. 2⇥ 1 chemical potential µ˜i µi Ui. This fact was emphasized in Sawyerq = (1975, 1989)< Burrows and Sawyer [2], and used to show that it was un- Iwamoto & Pethick (1982)R necessary to explicitly know the values of the nucleon poten- Horowitz & Wherberger (1991)tials for a given nuclear equation of state (which are often not In the following we show that the mean2 field energy shift,2 R Burrows & Sawyer (1999) easily available from widely used nuclear equations of state driven by the nuclear symmetry energy,⇤ = has a> similar but Reddy, et al. (1998,1999,2000,2001,2002)in the core-collapse supernova community) when calculating substantially larger effect in neutron-rich matter⇥ at densities⇥collision ⌧collision =12Collision3 Time the neutral current response of the nuclear medium. Clearly, if • Neutrinos⇤ 10 “see”g/cm more. than one particle2 in the medium.2 ⇤ = < both µi and µ˜i are known, then Ui can be easily obtained. This implies that for a given temperature, density and electron frac- ⇥ ⇥collision tion, the neutral current response function is unchanged in the • Nature of spatial andB. temporal Mean Field correlations Effects between nuclei, nucleons and electrons affect the scattering rate. presence of mean field effects, as the kinematics of the reaction are unaffected by a constant offset in the nucleon single Interactions in the medium alter the single particle energies, particle energies. In contrast, the kinematics of the charged • Nucleonand dispersion nuclear mean relation field theories is altered. predict a nucleonEnergy dispersion shifts are important.current reaction are affected by the difference between the relation of the form neutron and proton potential and the charged current response is altered in the presence of mean field effects. E (k)=⇤k2 + M 2 +U K(k) +U , (12) i ⇤ i i Inspecting the response function in Eq. 4 and the dispersion relation in Eq. 11 it is easily seen that the mean field response where M⇤ is the nucleon effective mass and Ui is the mean • Phase fieldtransitions energy shift. to Forquark neutron-rich matter, conditions, matter thewith neutron strangeness po- can greatly M2T exp (e˜ µ˜ )/T + 1 tential energy is larger due the iso-vector nature of the strong , min 2 , alter the neutrino mean free path. SMF(q0 q)= z ln ⇥q(1 e ) exp (e˜min µ˜2)/T + exp[z] interactions. The difference U =Un Up is directly related to ⇤ ⇥ ⌅ (13) the nuclear symmetry energy, which is the difference between ⇥ the energy per nucleon in neutron matter and symmetric nu- where clear matter. Ab-intio methods using Quantum Monte Carlo M 2 2 reported in [20] and [21], and chiral effective theory calcu- e˜min = (q0 +U2 U4 q /2M) , (14) 2q2 lations of neutron matter by [22] suggest that the symmetry energy at sub-nuclear density is larger than predicted by many is obtained from the free gas response by the replacements mean field models currently employed in supernova and neutron star studies (for a review see [23]). To highlight its im- µi µ˜i = µi Ui ⇥ portance we choose two models for the dense matter equation q0 q˜0 = q0 +U2 U4 (15) of state: (i) the GM3 relativistic mean field theory parameter ⇥ set without hyperons [24] where the symmetry energy is linear and q q. Therefore, we see that the potential difference at low density; and (ii) the IU-FSU parameter set [25] where U = ⇥(U U ) affects reaction kinematics and cannot be ± 2 4 Neutrino Rates

ν ν emission ω,q ω =q ω scattering

q d 2σ E ≈ G2 Im L (k,k + q) Πµν (q) V dcosθ dE $ F E $ [ µν ]

Lµν (k,k + q) = Tr [ lµ (k) lν (k + q) ] ! d 4 p µν ( ,q ) Tr [ j µ (p) jν (p q) ] Π ω = ∫ 4 + (2π) Dynamic structure factor: spectrum of density, spin and current fluctuations. € Neutrino Diffusion Enhanced by Nuclear Correlations ⇧µ⌫ (!,q)=

! ! p + q Γµ Γν = ! p € € € ! ! G (p + q ) Γµ µν Γν € € ! G (p) νµ € ! ! €+ ! € G (p + q ) ! µσ p + q Γµ € Γν Horowitz & Wehrberger, (1991) ! Burrows & Sawyer, (1999) ! Vστ p Reddy, Pons, Prakash, Lattimer, (1998,1999) Gσµ(p) € € € € € € € First-Order Phase Transitions in the PNS Core

exotic a A Mixed phase µ e = −µQ − − + b nuclear + € B

µ B

Mixed Phase: Heterogeneous phase with Nuclear Exotic structures€ of size 5-10 fm.

Reddy, Bertsch & Prakash, (2000) Neutrino Mean Free Path in a Mixed Phase

ν’

Scattering from quark droplets in a quark-hadron transition. 2 dσ GF 2 2 = ND Sq QW Eν(1 + cos(θ)) dcos (θ) 16π

Reddy, Bertsch & Prakash, (2000) € e + n p + e Charged Current Reactions ! + ⎨¯e + p n + e ! • Determine the electron neutrino spectra and deleptonization times.

• Final state electron blocking is strong for electron neutrino absorption reaction.

• Asymmetry between mean ﬁeld energy between neutrons and protons alters the kinematics.

⌫e e

q U = U U 0 ⇡ n p Reddy, Prakash & Lattimer (1998) Roberts (2012) n p Martinez-Pinedo et al. (2012) Roberts & Reddy (2012) SINGLE PARTICLE ENERGY SHIFT & DAMPING

p2 En(p) mn + + Un + i n ⇡ 2mn⇤ (p + q)2 Ep(p + q) mp + + Up + i p ⇡ 2mn⇤

Energy Transfer in the Charged Current Process:

pq q0 = En(p) Ep(p + q) +(mn mp)+(Un Up) ' 2mn⇤ 0 1.3MeV ⇡ '

nn np U = Un Up 40 MeV ⇡ n0 Figure 3. Elemental abundances in the halo star CS 22892-052 are compared with solar 1.00 system abundances attributable to the r- Pt process. The numerical values of the halo- Zr Figure 3. Elemental abundances0.50 in the halo star abundances follow the convention of Sn 1.00 Os star CS 22892-052 are compared with solarRu Ba figure 2. The solar system r-process abun- Ga Nd Dy system abundances attributableSr to the r-Cd Pt dances are scaled down to compensate for 0.00 Gd Zr the higher metallicity of the much process.younger The numerical values of the halo- 0.50 Er Pb star abundances follow the convention of Ce Sm Sn Os Sun. (Adapted from ref. 9.) Ba figure 2. The solar system r-process abun- Ga Yb RuIr Cd Nd Dy dances are scaled down–0.50 to Gecompensate for Sr Gd Y 0.00 Hf Er Pb the higher metallicity of the much younger Ce Sm Sun. (Adapted from ref. 9.) AL ABUNDANCE Yb Ir only over a very small range. Perhaps that –1.00 Mo La–0.50 Ge means that only a small minority of type II su- Ag Ho Y Hf pernovae, confined to a narrow mass range, Nb Pr Eu AL ABUNDANCE Au produce r-process elements. only over a very small–1.50 range. Perhaps that –1.00 Tb La ELEMENT Tm Mo Although abundance data means for specific that only a small minority of type II su- Ag Ho Lu PrTh isotopes in halo stars are much harderpernovae, to ac- confined to a narrow massHalo range,star Nb Eu Au quire than the spectroscopic dataproduce that pro-r-process elements.–2.00 Upper limits –1.50 Tb ELEMENT Tm vide the elemental abundances of figureAlthough 3, abundance data for specific U Solar system r-process Th recent isotopic observations appearisotopes to be in in halo stars are much harder to ac- Halo star Lu quire than the spectroscopic–2.50 data that pro- –2.00 Upper limits agreement with the elemental abundance 30 40 50 60 70 80 90 trends. In particular, it has beenvide found the that elemental abundances of figure 3, Solar system r-process U recent isotopic observations appear to be in ATOMIC NUMBER Z the two stable isotopes of europium are –2.50 found in the same proportion in severalagreement old, with the elemental abundance 30 40 50 60 70 80 90 metal-poor halo stars as theytrends. occur In in particular, solar system it has been150 found neutrons that per seed nucleus. Iron is generallyATOMIC the NUMBER light- Z r-process material.11 the two stable isotopes of europiumest of the arerelevant seed nuclei. Modelers of r-process nu- found in the same proportion in several old, That is not particularly surprising, because Eu is still cleosynthesis find the entropy of the expanding matter and metal-poor halo stars as they occur in solar system 150 neutrons per seed nucleus. Iron is generally the light- synthesized overwhelmingly by the r-process. But what the overall neutron/proton ratio to be more useful param- r-process material.11 est of the relevant seed nuclei. Modelers of r-process nu- about elements like Ba that, unlike Eu, are nowadays pri- That is not particularly surprising,eters than because temperature Eu is still and neutroncleosynthesis density. find In the a veryentropy neu- of the expanding matter and marily made by the s-process? A recent study has found synthesized overwhelmingly bytron-rich the r-process. environment But what such asthe a neutronoverall neutron/proton star, the r-process ratio to be more useful param- that the relative abundance of different Ba isotopes in one 8 about elements like Ba that, unlikecould Eu, occur are evennowadays at low pri- entropy.eters thanBut eventemperature a small and excess neutron density. In a very neu- very old halo star is compatible with the Ba isotope ratio marily made by the s-process?of A neutrons recent study over has protons found cantron-rich sustain environment the r-process such if asthe a neutron star, the r-process 12 14 attributable to the r-process in that solar the system relative material. abundance of entropydifferent isBa high isotopes enough. in one could occur even at low entropy.8 But even a small excess The Eu and Ba isotope results supportvery old the halo conclusion star is compatible that withThe the question Ba isotope is, Whereratio inof natureneutrons does over one protons find the can ap- sustain the r-process if the only the r-process was producingattributable heavy elements to the r-process in the inpropriate solar system conditions material.—either12 entropyvery neutron-rich is high enough. material14 at early galaxy. The Eu and Ba isotope results lowsupport entropies the conclusion or moderately that neutron-richThe question material is, Where at high in nature does one find the ap- Elemental abundance patternsonly the from r-process additional was producing entropies? heavy elementsBut if the in entropy the propriate is too high, conditions there —willeither be toovery neutron-rich material at r-process-rich halo stars now addearly support galaxy. to this conclu- few seed nuclei to initiate the r-process. The extreme case 3 low entropies or moderately neutron-rich material at high sion. All the stars in this sample haveElemental Eu/Fe abundance abundance patternsis the Big fromBang, additional from which entropies?4He was essentially But if the entropythe heav- is too high, there will be too ratios that typically exceed that r-process-richof the Sun by halo at least stars an now add support to this conclu- few seed nuclei to initiate the r-process. The extreme case 3 iest surviving nucleus. order of magnitude. Much less work,sion. All however, the stars has in this been sample haveDetermining Eu/Fe abundance whether r-processis the Big conditions Bang, from canwhich occur 4He was essentially the heav- done on r-process-poor halo stars.ratios The thathalo typically stars presum- exceed thatinside of the type Sun II by supernovae at least an requiresiest surviving an understanding nucleus. of the order of magnitude. Much less work, however, has been ably got their heavy elements from material spewed out nature of those stellar catastrophes.Determining The most whether plausible r-process conditions can occur done on r-process-poor halo stars. The halo stars presum- by supernova explosions of an even earlier generation of mechanism for such an explosioninside oftype a massiveII supernovae star requiresis en- an understanding of the massive, short-lived stars. So notably all got halo their stars heavy acquired elements from material spewed out ergy deposition in the star’snature outer precincts of those stellar by neutrinos catastrophes. The most plausible the same share of these r-processby ejecta. supernova In halo explosions stars poor of an even earlier generation of mechanism for such an explosion of a massive star is en- massive, short-lived stars. So streamingnot all halo from stars the acquired hot proto-neutron star formed by the in r-process elements, the heavy elements are much harder ergy deposition in the star’s outer precincts by neutrinos the same share of these r-processgravitational ejecta. In halo collapse stars poor of the central iron-core when all the to identify spectroscopically. But studies of those very stars streaming from the hot proto-neutron star formed by the in r-process elements, the heavyfusion elements fuel are is exhaustedmuch harder (see figure 4). The dominant neu- might provide important clues about their massive pro- gravitational collapse of the central iron-core when all the to identify spectroscopically. Buttrino studies energy of those deposition veryYe in stars the processes Neutrino Drivenare Wind genitors—the galaxy’s first stars. fusion fuel is exhausted (see figure 4). The dominant neu- might provide important clues about their massive pro- Figure 3 also shows that the abundances of the lighter n +nO p+e– and n+ +pO n+e+. genitors—the galaxy’s first stars. Ise set by the charged currenttrino energye deposition processes are n-capture elements, from Z = 40–50, generally fall below reactions in two regions. { – + Figure 3 also shows that theThe abundances neutrino ofheating the lighter efficiency dependsne +n onO p+econvectiveand in- n+e+pO n+e. the r-process curve that fits the heavier elements so well. ˙ n-capture elements, from Z = stabilities40–50, generally NDW and the fall opacitybelowN⌫e ⌫e of the stellar material to the That difference is suggestive. It might be telling us that Ye h i The neutrino heating efficiency depends on convective in- the r-process curve that fits the heavier elements⇡ N˙ so well.+ N˙ the r-process sites for the lighter and heavier n-capture el- transit of neutrinos.⌫¯e h ⌫¯ Thee i actual⌫e stabilitiesh ⌫e i explosion and the mechanism opacity of is the stellar material to the That difference is suggestive. It might be telling7,14,15 us that 13 still uncertain. 2 Self-consistent2 supernova calculations ements are somehow different. Possible alternative sites ⌫¯ E ⌫ Etransit of neutrinos. The actual explosion mechanism is the r-process sites for the lighter and heaviere n-capture⌫¯e eel- ⌫e with presentlyh i/h knowni h neutrinoi/h i physics have7,14,15 not yet pro- for the r-process include neutron-starements binaries are somehow as well different. as 13 Possible alternative sites still uncertain. Self-consistent supernova calculations duced successfulNeutrino-sphere explosions. at high density supernovae, or perhaps just differentfor the astrophysical r-process include condi- neutron-star binaries as well as with presently known neutrino physics have not yet pro- tions in different regions of a single core-collapse super- Thereand moderateis hope, entropy. however, duced that thesuccessful neutrino-driven explosions. ex- supernovae, or perhaps just different astrophysicalR ~ 10-20 km condi- 3 plosion mechanism will prove to be right when the effects nova. Further complicating the interpretation,tions in different strontium, regions of a single core-collapse super- There is hope,PNS however, that the neutrino-driven ex- yttrium, and zirconium (Z = 38–nova.40) 3seemFurther to havecomplicating a very theof interpretation, stellarNeutrino rotation driven strontium, wind and at magneticlow-plosion fieldsmechanism are included will prove in to be right when the effects complex synthesis history that raises the specter of multi- model calculationsdensity and high that entropy. are notof restricted stellar rotation to spherical and magnetic sym- fields are included in yttrium, and zirconium (Z = 38–40) seem to3 have4 a very ple r-processes. complex synthesis history thatmetry. raises theThereR ~ specter10 -10is also km of multi- still muchmodel uncertainty calculations in that our are knowl- not restricted to spherical sym- ple r-processes. edge of how neutrinos interactmetry. with There dense is matter also still (and much in- uncertainty in our knowl- Is it always supernovae? deed of how they behaveedge in of vacuum). how neutrinos The interact lack of with dense matter (and in- The critical parameter that determinesIs it always whethersupernovae? the understanding of the type IIdeed supernova of how explosion they behave mecha- in vacuum). The lack of r-process occurs is the number ofThe neutrons critical per parameter seed nu- thatnism determines also means whether that we the do understandingnot know the ofexact the typer-process II supernova explosion mecha- cleus. To synthesize nuclei with Ar-processabove 200 occurs requires is the about numberyields of neutrons for these per supernovae. seed nu- nism also means that we do not know the exact r-process cleus. To synthesize nuclei with A above 200 requires about yields for these supernovae. 50 October 2004 Physics Today http://www.physicstoday.org 50 October 2004 Physics Today http://www.physicstoday.org Spectra & Nucleosynthesis

νe

νe w/o 12 nuclear effects > (MeV)

ν with nuclear effects ε <

0.6 proton-rich 0.55 nuclear effects w/o ejecta

0.5 e,NDW

Y neutron-rich ejecta Still EoS 0.45

0.4 0246 8 10 Soft EoS Time (s) Figure 3.5: Top panel:FirstenergymomentoftheoutgoingelectronneutrinoandaRoberts,ntineutrino Reddy & Shen (2012) as a function of time in three PNS cooling simulations. The solid lines are the average energies of the electron neutrinos and the dashed lines are for electron antineutrinos. The black lines correspond to a model which employed the GM3 equation of state, the red lines to a model which employed the IU-FSU equation of state, and the green lines to amodelwhichignoredmeanﬁeld eﬀects on the neutrino opacities (but used the GM3 equation ofstate). Bottom panel:Predicted neutrino driven wind electron fraction as a function of time for the three models shown in the top panel (solid lines), as well as two models with the bremsstrahlung rate reduced by a factor of four (dot-dashed lines). The colors are the same as in the top panel.

66 Spectra at late times

50 2×10 20

50 2×10 U 10 ) (MeV) -1

MeV 5 T

-1 50 1×10 (erg s e 0 L

-2 49 5×10 10

) 50

-3 2×10 -3 (fm 10 B 0 n 0 10 20 30 40 50 (MeV) 2×10 -4 10 ) -1 0.12 MeV

-1 50 • Decoupling occurs at 1×10 (erg s e 0.08 relatively high density. L e Y 49 5×10 0.04 • Spectra inﬂuenced by

0 energy shifts and nuclear 00 10 20 30 40 10 20 30 40 50 (MeV) (MeV) correlations. Figures from PNS simulations by Roberts (2012) Spectra at late times

50 2×10 20

50 2×10 U 10 ) (MeV) -1

MeV 5 T

-1 50 1×10 (erg s e 0 L

-2 49 5×10 10

) 50

-3 2×10 -3 (fm 10 B 0 n 0 10 20 30 40 50 (MeV) 2×10 -4 10 ) -1 0.12 MeV

-1 50 • Decoupling occurs at 1×10 (erg s e 0.08 relatively high density. L e Y 49 5×10 0.04 • Spectra inﬂuenced by

0 energy shifts and nuclear 00 10 20 30 40 10 20 30 40 50 (MeV) (MeV) correlations. Figures from PNS simulations by Roberts (2012) PNS evolution: Role of EoS. Heat transport : Neutrino diffusion + convection Diffusion: R2 ⌧di↵ 3 5s ' c⌫ ⇡ convection

Convection: neutrino diffusion Convection is driven by composition and entropy gradients.

The buoyancy of matter depends nuclear equation of state. Observable signatures of convective transport 3

6 No Convection g’=0.6 GM3 0.1 0.1 LargeConvection L MF GM3 5 •Neutrino ﬂux is w/o convection Convection g’=0.6 GM3 Convection(stiﬀ) g’=0.6 IU-FSU 4 enhanced during 3 10 1 1 3 convection. Small L Entropy ) ) 1 ∞ − (soft)

2 >

5 5 − 0.35 10 There is break in the 1 15 •10 15 30 30 light curve (when 2 0.3 0.4 10 Count Rate (s 0.35 convection ends). 0.1 0.1 0.25 > 10 s)/ Counts (0.1 s 0.3 − 1 1 0.25 0.2 5 •Fraction of events L 5 Y

0.2 Counts (3 s 10 15 10 between 3-10 s 1 0.3 0.35 0.4 0.45 0.15 30 10 Counts (0.1 s −> 1 s)/ Counts (0.1 s −> ) 15 ∞ 0.1 provides good 30 0 1 10 10 0.05 discrimination. Time (s) 0 0 0.5 1 1.5 0 0.5 1 1.5 Baryon Number (solar units) Baryon Number (Solar Units) FIG. 3:Count Count rate rates in as Super-Kamiokande a function of time for for a number of 1.6M PNSgalactic models. supernova Inset: The at 10 integrated kpc. number of counts ⊙ FIG. 2: Evolution of the entropy (top panels) and lepton frac- from 100 ms to 1 s divided by the total number of counts tion (bottom panels) in a 1.6M rest mass PNS for the GM3 ⊙ for t>0.1secondonthehorizontalaxis,andthenumberofRoberts, Cirigliano, Pons, Reddy, Shen, Woosley (2012) EoS (left panels) and the IU-FSU EoS (right panels). The counts for t>3secondsdividedbythetotalnumberofcounts grayed regions are convectively unstable. The labels corre- for t>0.1second.Symbolsizescorrespondtovariousneu- spond to the model times in seconds. tron star masses ranging from 1.2M to 2.1M .Colorscorre- ⊙ ⊙ spond to different values of the Migdal parameter increasing as the colors lighten (g′ =0.2, 0.6, 1.0), with the black points path. To take this into account, we introduce an effec- being mean field results. The circles correspond to the GM3 tive short-range interaction in the spin channel through EoS and the stars to the IU-FSU EoS. the Migdal parameter, g′ [18]. The strength of this interaction is tuned to reproduce the spin-suscpetibitly of neutron matter obtained from microscopic calculations [7]. IU-FSU. This difference can bedirectlyattributedtothe For densities above nuclear saturation, the RPA causes a difference in Esym′ between the two EoSs. As the mantle significant enhancement of the mean free path relative to contracts, the second term in equation 2 becomes increas- the mean field approximation due to the repulsive nature ingly dominant and is eventually able to stabilize convec- of the nuclear interaction at high density. This should be tion. The exact details of how convection proceeds will considered as only a first approximation to the actual depend on the initial conditions of the PNS. Still, qual- response of the nuclear medium, as we only include sin- itatively, increasing Esym′ will shut-off convection at an gle particle-hole excitations and it has been shown that earlier time. multi-particle-hole excitations may be important in de- The depth to which convection penetrates in the core termining the axial current component of the neutrino and how long convection proceeds in the core is depen- opacity [21]. dent upon the opacities as well as the EoS. When only We now consider the evolution of the internal structure mean field effects on the opacities are considered (i.e. of the PNS with convection and varying prescriptions for when the neutrino mean free path is shorter), convection the opacities. In figure 2, the evolution of the entropy does not proceed all the way to the center of the PNS in and electron fraction for the two equations of state are the GM3 models. When RPA effects are included, con- shown. Over the first second in both models, convection vection does proceed to the central regions of the core. smoothes the entropy and lepton gradients in the outer An increased diffusion rate allows the core to heat up and regions to a state close to neutral buoyancy. GM3 has deleptonize more rapidly, thereby decreasing the stabi- aslightlysteeperentropygradient because of its larger lizing lepton gradients and increasing the de-stabilizing entropy gradients. Esym′ than IU-FSU. This results in a slightly larger neutrino luminosity at early times for GM3. As time pro- Of course, variations in the convective evolution of the gresses, convection steadily digs deeper into the core of PNS are only interesting to the extent they are poten- the PNS. For both EoSs, convection proceeds all the way tially observable in the neutrino emission from a nearby to the core by 15 seconds into the simulation, but it lasts supernova. In figure 3 the expected neutrino count rates in the interior regions for a much longer period of time for a detector similar to Super Kamiokande-III are shown for IU-FSU resulting in more rapid lepton depletion in for a number of PNS cooling models. We have assumed the core. More important to the neutrino signal, in GM3 athresholdenergyof7.5MeV,adetectormassof50kt, convection ceases in the mantle by 5seconds,whereas adetectorefficiency above threshold of unity [14], and a convection in the mantle proceeds until∼ 12 seconds in distance of 10 kpc to the supernova. Equipartition has ∼ over uncertain parameters, as well as the derivation of a bound with a clear probabilistic interpretation and unambiguously-defined confidence levels. Our results, and a discussion of them, are presented in 4. §

2PNSEvolutionwithAdditionalSourcesofEnergyLossandSN 1987a signal KII IMB KII 12 IMB 12 Hypothetical WeaklyCore-collapse Interacting supernovae are incredibly Particles rich physical sys tems, which include 10 10 many complicated phenomena. Indeed there is still much debate about basic Since neutrinos are trapped for ~20features s likea new the details particle of the explosion that mechanismcouples [22,23 ]. What is, how- 8 8 more weakly to matter can radiateever, away now well-established the binding is that theenergy birth and. subsequent evolution of the 6 PNS located in the inner region of the supernova provides the main source

N(t) 6 of neutrinos during the first several secondsN(t) after core bounce. Nearly all of 19 When this energy loss > 10 ergs/g/s4 the binding it energy will gained shorten during thethe collapse neutrino is trapped time inside the PNS and this energy is radiated by neutrino diffusion4 over several tens of seconds [8– scale by a factor of 2. 2 11]. These time scales and the associated neutrino luminosities are affected by several pieces of physics, including the2 total mass of the PNS, and both 0 KII IMB the0 nuclear 5 equation 10 of state0 (EOS) and 5 neutrino 10 opacities 15 atsupra-nuclear density. None12 of these quantities can be0 determined in a model-independent t [s] Hanhart, Pons, Reddy,0t [s] Phillips 5 (2001) 10 0 5 10 15 approach, but earlier studies have explored the model sensitivities by using t [s] t [s] different EOSs and the associated self-consistently-calculated opacities [11]. Raffelt (1996) Fig. 1. ComparisonThese of the studies results10 indicate of our that simulations the critical for physical a PNS paramete with arthatdetermines baryonic mass of M =1.6 M⊙ toneutrino the data luminosity fromFig. SN in 1. 1987a. theComparison first N(t) several denotes seconds of the the is results the integrated PNS of total our number mass. simulations Theof for a PNS with a baryonic mass counts. We show resultsuncertainties from a associated model without with the graviton properties emissi of denseon hadr (solidonic line), matter as do well not of8 M =1.6 M⊙ to the data from SN 1987a. N(t) denotes the integrated number of as the results witha gravitonffect these emission luminosities for greatly,a2 =0 provided.01 MeV/baryon/s that the opacities (short-dashed are calculated consistently withcounts. the EOS, We because show important results from feedbacks a model between without the EOS graviton emission (solid line), as well line) and a2 =0.1 MeV/baryon/s (long-dashed line). and the opacities6 tend to reduce the differences. The protoneutron star evolu-

Axions N(t) as the results with graviton emission for a2 =0.01 MeV/baryon/s (short-dashed tion code employed for this study has been described in detailinRefs.[11,12]. inition. However, itDetails must regarding be borneline) the neutrino and in minda2 opacities=0 that.1 employedtheMeV/baryon/s neutrino can be fomundean-free (long-dashed in Ref. [24]. paths In line). are energy dependent,arecentpapertheratesfortheemissivitiesofKK-graviton and thus4 the concept of a neutrinospheresindensematter is only approximate. In thewere next computed sectioninition. in we a model will However, independent study the way it eff must usingect of low-energy be thos borneeuncertainties theorems in mind which that the neutrino mean-free paths relate the emissivitiesneutrino to the well temperature measured nucleon-nucleon cross sections [4]. related to this definitionIn a world of with theare2n =2or energyKaluza-Kleinn =3compactGODstheKK-gravitonemissivity dependent, Gravitonsby exploringand thus the the sensitiv- concept of a neutrinosphere is only ap- ity of the count ratesof neutron-star and theproximate. deduced matter can bounds be fitted In the to to within changes next 5% section accuracy in our weprescription. overarangeof will study the effect of thoseuncertainties In Fig. 1 we showtemperatures the number from0 of 15 accumulated to 30 MeV and a counts range of as densities a functio from 0noftime,.5n0 to 3n0, related−3 to this definition of the neutrino temperature by exploring the sensitiv- N(t), in Kamioka (leftwhere panel)n0 =0.16 and fm0 IMBis the (right nuclear 5 panel). saturation The 10 density. results Theshown0 result, are which for 5 10 15 includes contributionsity of from thenn count, pp,and ratesnp collisions and the is: deduced bounds to changes in our prescription. abaryonicmassof1.6M⊙ (with a final gravitationalt [s] mass of 1.46 M⊙), and t [s]

In Fig. 1 we showpn the number of accumulated counts as a functionoftime, an anti-neutrino temperaturedE definednB as above.T They include the case without = an χ(Xn,Xp)MeV/baryon/s(1) KK-graviton emission, anddt twoN(t cases),n in0 Kamiokawith10 MeV radiation (left to panel) the GO andDs IMB included, (right panel). The results shown are for Fig. 1. Comparison of! the"! results" of our simulations for a PNS with a baryonic mass namely: a2 =0.01 MeV/baryon/sabaryonicmassof1.6M and a2 =0.1MeV/baryon/s.Asarguedin⊙ (with a final gravitational mass of 1.46 M⊙), and the introduction,of M =1 KK-gravitons.6 M⊙ toan the steal anti-neutrino data energy from from4 temperature the SN core 1987a. and defined thereby N(t) as dra- above. denotes They the include integratedthe case number without of maticallycounts. suppress We the show late-time resultsKK-graviton neutrino from emission. emission, a model Note and without that two the cases early-time graviton with radiation emission to the (solid GODs line), included, as well neutrino emission is not strongly influenced by the existenceofGODsbecause namely: a2 =0.01 MeV/baryon/s and a2 =0.1MeV/baryon/s.Asarguedin theseas neutrinos the results are emitted with from graviton the lower density emission and lower fortemperaturea2 =0. re-01 MeV/baryon/s (short-dashed the introduction, KK-gravitons steal energy from the core and thereby dra- gionsline) of the starand wherea2 =0 the graviton.1 MeV/baryon/s emissivity is small. (long-dashed However, after line).about 2–4 seconds, neutrino losses provokematically a strong suppress compression the late-time and heating neutrino of the emission. Note that the early-time star [8,10,11], which activatesneutrino the KK-graviton emission emissivit is not stronglyy. Consequently, influenced the by the existenceofGODsbecause totalinition. number of neutrinoHowever, counts,these it which must neutrinos clearly be are borne depends emitted on in th from mindetotalamount the that lower the density neutrino and lower mtemperatureean-free paths re- of energyare radiated energy in neutrinos, dependent,gions drops of asthe anda2 staris increased. thus where the the graviton concept emissivity of a neutrinosphe is small. However,re after is only about approximate. In the2–4 next seconds, section neutrino we losses will provoke study a strong the eff compressionect of thos andeuncertainties heating of the This total number of countsstar also [8,10,11], has a significant which dependence activates theon KK-graviton the PNS emissivity. Consequently, the mass.related Indeed, the to mass this of the definition PNS is generally of the the mostneutrino important temperature factor in the by exploring the sensitiv- neutrino signal [11]. However,total it is worth number noting of that, neutrino if a suffi counts,ciently importantwhich clearly depends on thetotalamount additionality of sink the of energy count is included ratesof energy in and the radiated PNS the simulation, deduced in neutrinos,the time-structure bounds drops as toa2 is changes increased. in our prescription. of theIn signal Fig. actually 1 we becomes show less the sensitive number to the precise of accumulatedvalue of the PNS counts as a functionoftime, mass. This happens becauseThis more massivetotal number stars attain of counts higher alsotemperatures, has a significant dependence on the PNS whichN in(t turn), in results Kamioka in enhancedmass. (left graviton Indeed, panel) emission. the andmass Conseq of IMB theuently, PNS (right is much generally panel). of the The most results importantshown factor in are the for abaryonicmassof1.6Mneutrino signal⊙ [11].(with However, a final it is gravitational worth noting that, mass if a suffi ofciently 1.46 important M⊙), and additional sink of energy is included in the PNS simulation, the time-structure an anti-neutrino temperature6 defined as above. They include the case without of the signal actually becomes less sensitive to the precise value of the PNS KK-graviton emission,mass. This and happens two because cases more with massive radiation stars attain to the higher GOtemperatures,Ds included, namely: a2 =0.01which MeV/baryon/s in turn results in and enhanceda2 =0 graviton.1MeV/baryon/s.Asarguedin emission. Consequently, much of the introduction, KK-gravitons steal energy from the core and thereby dra- matically suppress the late-time neutrino emission.6 Note that the early-time neutrino emission is not strongly influenced by the existenceofGODsbecause these neutrinos are emitted from the lower density and lower temperature regions of the star where the graviton emissivity is small. However, after about 2–4 seconds, neutrino losses provoke a strong compression and heating of the star [8,10,11], which activates the KK-graviton emissivity. Consequently, the total number of neutrino counts, which clearly depends on thetotalamount of energy radiated in neutrinos, drops as a2 is increased.

This total number of counts also has a signiﬁcant dependence on the PNS mass. Indeed, the mass of the PNS is generally the most important factor in the neutrino signal [11]. However, it is worth noting that, if a suﬃciently important additional sink of energy is included in the PNS simulation, the time-structure of the signal actually becomes less sensitive to the precise value of the PNS mass. This happens because more massive stars attain higher temperatures, which in turn results in enhanced graviton emission. Consequently, much of

6 over uncertain parameters, as well as the derivation of a bound with a clear probabilistic interpretation and unambiguously-deﬁned conﬁdence levels. Our results, and a discussion of them, are presented in 4. §

Axions N(t) as the results with graviton emission for a2 =0.01 MeV/baryon/s (short-dashed tion code employed for this study has been described in detailinRefs.[11,12]. inition. However, itDetails must regarding be borneline) the neutrino and in minda2 opacities=0 that.1 employedtheMeV/baryon/s neutrino can be fomundean-free (long-dashed in Ref. [24]. paths In line). are energy dependent,arecentpapertheratesfortheemissivitiesofKK-graviton and thus4 the concept of a neutrinospheresindensematter is only approximate. In thewere next computed sectioninition. in we a model will However, independent study the way it eff must usingect of low-energy be thos borneeuncertainties theorems in mind which that the neutrino mean-free paths relate the emissivitiesneutrino to the well temperature measured nucleon-nucleon cross sections [4]. related to this definitionIn a world of with theare2n =2or energyKaluza-Kleinn =3compactGODstheKK-gravitonemissivity dependent, Gravitonsby exploringand thus the the sensitiv- concept of a neutrinosphere is only ap- ity of the count ratesof neutron-star and theproximate. deduced matter can bounds be fitted In the to to within changes next 5% section accuracy in our weprescription. overarangeof will study the effect of thoseuncertainties Ruled OutIn Fig. 1 we showtemperatures the number from0 of 15 accumulated to 30 MeV and a counts range of as densities a functio from 0noftime,.5n0 to 3n0, related−3 to this definition of the neutrino temperature by exploring the sensitiv- N(t), in Kamioka (leftwhere panel)n0 =0.16 and fm0 IMBis the (right nuclear 5 panel). saturation The 10 density. results Theshown0 result, are which for 5 10 15 includes contributionsity of from thenn count, pp,and ratesnp collisions and the is: deduced bounds to changes in our prescription. abaryonicmassof1.6M⊙ (with a final gravitationalt [s] mass of 1.46 M⊙), and t [s]

2MadappaPrakashetal.

(important in determining a neutron star’s maximum mass), symmetry energies (important in determining the typical stellar radius and in the relative proton fraction) and specific heats (important in determining the local temperature). These characteristics play important roles in determining the matter’s composition, in particular the possible presence of additional components (such as hyperons, a pion or kaon condensate, or quark matter), and also significantly affect calculated neutrino opacities and diffusion time scales. The evolution of a PNS proceeds through several distinct stages [1,2] and with various outcomes, as shown schematically in Fig. 1. Immediately following core bounce and the passage of a shock through the outer PNS’s mantle, the star contains an unshocked, low entropy core of mass Mc 0.7M⊙ in which neutrinos are trapped (the first schematic illustration, labelled≃ (1) in the figure). The core is surrounded by a low density, high entropy (5

Prakash, Lattimer, Pons, Steiner and Reddy, astro-ph/0012136 Fig. 1. The main stages of evolution of a neutron star. Shading indicates approximate relative temperatures.

After a few seconds (stage 2), accretion becomes less important if the supernova is successful and the shock lifts oﬀ the stellar envelope. Extensive neutrino